Hi Vivanco! Welcome to cruise chat!
This question pops up often. Yes, you can save a lot of money (especially with a family as large as yours) by booking excursions on your own.
BUT doing so you take on the risk of missing the ship because of time difference and or transportation issues with your tour operator.
It also depends on the port. There are some ports we don't recommend you going off on your own on, but there are many that are very easy to do this in. It would help if you provided us a list of the ports you are going to stop at and we can then tell you what your options are.
There are some tours you should book through the cruise line. These would be any that have significant travel involved. example- Going to the Mayan ruins from Cozumel. These involve taking a ferry to the mainland and then a long bus ride. Not easy to do on your own AND you are taking a big risk in getting back to the ship on time. Then there are islands where if you are on a tour where their bus breaks down you could easily get a taxi back to the ship.
If you provide us with your itinerary there are many very experience cruise chat members that can help you book tours on your own. It would also help if you give us an idea of what your interests are.
Carnival Cruise Lines:
Fantasy 90, Celebration 95, Victory 05, Conquest 05, Spirit 6/06, Freedom 6/08, Holiday 7/08, Victory 6/09, Valor 9/09, Dream 6/10, Splendor 8/10,Conquest 8/11,Magic 11/11,Triumph 4/12,Spirit 7/12,Freedom 11/12, Triumph 2/13 - The fire cruise
NCL Dream October 2006, RCCL Mariner OTS June 2007, Celebrity Silhouette 2013